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PLANE  CURVES  OF  THE  EIGHTH  ORDER 

WITH  TWO  REAL  FOUR-FOLD  POINTS 
HAVING  DISTINCT  TANGENTS  AND 
WITH  NO  OTHER  POINT  SINGULARITIES 


BY 


ELIZABETH   BUCHANAN   COWLEY 


Submitted  in   Partial   Fulfilment  of   the   Requirements  for 

THE  Degree  of  Doctor  of  Philosophy,  in  the  Faculty 

OF  Pure  Science,  Columbia  University 


PRESS  Op 

The  New  Era  Printing  CoMPAur 
Lancaster.  Pa. 

1908 


^■**.* 


PLANE  CURVES  OF  THE  EIGHTH  ORDER 

WITH  TWO  REAL  FOUR-FOLD  POINTS 
HAVING  DISTINCT  TANGENTS  AND 
W^ITH  NO  OTHER  POINT  SINGULARITIES 


BY 

ELIZABETH    BUCHANAN   COWLEY 


Submitted  in   Partial    Fulfilment  of   the   Requirements  for 

THE  Degree  of  Doctor  of  Philosophy,  in  the  Faculty 

OF  Pure  Science,  Columbia  University 


OF  THE     " 

UNIVERsn 

OF 


Press  ok 

The  New  Era  Printing  compan- 

Lancaster.  Pa 

190.S 


PLANE  CURVES  OF  THE  EIGHTH  ORDER 

With  Two  Real  Four-fold  Points  Having  Distinct  Tan- 
gents AND  with  No  Other  Point  Singularities. 

The  purpose  of  this  paper  is  the  discussion  of  plane  curves  of  the 
eighth  order  with  two  four-fold  points  having  distinct  tangents  and 
with  no  other  point  singularities.*  The  work  can  be  conveniently 
arranged  under  four  heads  : 

1 .  Possible  forms  of  curves. 

2.  Existence. 

3.  Classification. 

4.  Relation  to  space  curves. 

1.  Possible  Forms  of  Curves. 

These  plane  curves  of  eighth  order  C^  with  two  four-fold  points 
0  and  0'  are  of  deficiency  nine.  Hence  the  number  of  branches 
cannot  exceed  ten.f  C^  can  cut  the  line  00'  at  no  points  except  0 
and  0' .  If  there  are  odd  branches,  there  must  be  an  even  number 
of  them,  since  the  curve  is  of  even  order.  Each  odd  branch  passes 
an  odd  number  of  times  through  one  0  and  an  even  number  of  times 
through  the  other  0,  for  each  odd  branch  cuts  a  line  in  an  odd  num- 
ber of  points.  All  odd  branches  must  pass  an  odd  number  of  times 
through  the  same  0  (say  0) ;  for  otherwise  two  odd  branches  would 
intersect  in  an  even  number  of  points.  As  odd  branches  may  occur, 
it  is  necessary  to  distinguish  between  finite  and  infinite  for  the  pres- 
ent at  least.  The  curves  will  be  considered  as  they  appear  when 
projected  so  as  to  cut  the  line  at  infinity  the  least  possible  number  of 
times.  For  convenience,  the  term  loop  will  be  used  to  denote  a  part 
of  a  circuit  that  starts  from  one  0  and  returns  to  the  same  0  with- 
out passing  through  the  other  0 ;  and  the  term  intermediate  part  for 
a  portion  extending  from  one  0  to  the  other.  No  intermediate  part 
can  be  a  complete  circuit ;  a  loop  may  or  may  not  be  a  whole  branch. 
According  to  the  scheme  of  classification  which  is  adopted  (see  page 

*  No  papers  on  this  subject  are  known  to  me. 

t  Harnack,  Mathematische  Annalen,  X,  p.  189,  1876. 

3 


-j    <Qp    fl  r^jT^ 


4  PLANE    CURVES    OF    THE    EIGHTH    ORDER. 

9),  a  curve  which  lies  entirely  in  the  finite  portion  of  the  plane  is 
not  essentially  changed  if  some  of  its  loops  or  intermediate  parts  are 
stretched  out  to  cut  the  line  at  infinity  an  even  number  of  times. 
Hence  it  is  necessary  to  consider  only  those  cases  in  which  the  infi- 
nite loops  and  intermediate  parts  cut  the  line  at  infinity  only  once. 
Two  such  loops  must  intersect  once  at  least,  just  as  two  real  odd 
algebraic  curves  cut  in  at  least  one  real  point.  Hence  all  the  infinite 
loops  must  be  at  the  same  0,  for  if  they  do  not  cut  at  an  0  a  double 
point  is  introduced  into  the  C\ 

Let  a  =  the  number  of  infinite  loops  at  0. 
h  =  the  number  of  finite  loops  at  0. 
c  =  the  number  of  finite  loops  at  0', 
d  =  the  number  of  infinite  intermediate  parts. 
e  =  the  number  of  finite  intermediate  parts. 

Since  there  are  four  distinct  tangents  at  each  0,  there  are  eight 
directions  by  which  a  moving  point  may  leave  an  0.  As  an  inter- 
mediate part  absorbs  one  direction  at  each  0,  the  number  of  loops  at 
0  equals  the  number  at  0'.  Therefore,  a  +  b  =  c  (1).  The  number 
ez=S  —  (a-\-b-\-G  +  d)=  8  —  2(a  +  6)  —  d.  Consider  a  line 
which  cuts  each  of  these  finite  intermediate  parts  and  each  infinite 
loop  in  one  real  point ;  i.  e.,  a  line  cutting  line  00'  between  0  and  0'. 
The  number  of  real  points,  8  —  2(a  -f  6)  —  cZ  4-  a  is  either  equal  to 
or  greater  than  {a  +  d) ;  for  otherwise,  by  a  homographic  transfor- 
mation, one  could  reduce  the  number  of  real  points  in  which  the 
C^  is  cut  by  the  line  at  infinity.  This  last  condition  reduces 
to  4:  =  a  -\-  b  -\-  d  (2).  Moreover,  the  number  of  real  points  in 
which  the  line  at  infinity  cuts  C^  is  even.  Hence  a  +  cZ  is  an  even 
number  (3).  All  real  positive  values  of  a,  b,  c,  d,  e  which  satisfy 
these  three  equations  are  given  in  the  following  table.  The  number 
T  represents  the  number  of  types. 


a 

6 

c 

d 

e 

T 

A 

0 

0 

0 

0 

8 

1 

B 

1 

1 

6 

6 

C 

2 

o 

4 

30 

D 

3 

3 

• 

2 

21 

E 

4 

4 

0 

7 
65 

PLANE    CURVES   OF   THE    EIGHTH    ORDER. 
II         ^         (1)         0         0         0         2         6  1 


III 


(2) 

1 

1 

4 

12 

(3) 

2 

2 

2 

45 

B 

(1) 

1 

0 

1 

1 

5 

3 

(2) 

1 

2 

3 

30 

(3) 

3 

1 

54 

C 

(1) 

2 

0 

2 

0 

4 

2 

(2) 

1 

3 

2 

9 

(3) 

2 

4 

0 

15 
171 

A 

0 

0 

0 

4 

4 

1 

B 

1 

0 

1 

3 

3 

3 

C 

2 

0 

2 

2 

2 

2 

D 

3 

0 

3 

1 

1 

1 

E 

4 

0 

4 

0 

0 

0 

7 

Total  243 

Consider  class  I  in  which  all  the  curves  lie  entirely  in  the  finite 
portion  of  the  plane. 

A.  Numbering  the  eight  directions  from  each  0  as  in  figure  1  it 
is  readily  seen  that  the  only  way  to  get  the  eight  intermediate  parts 
without  introducing  double  points  is  by  joining  the  directions  having 
corresponding  numbers.  Hence  there  is  only  one  form  possible.  It 
has  four  branches.* 

B.  If  the  finite  loops  at  0  and  0'  were  on  different  sides  of  the 
line  00' ,  there  would  be  in  each  half  plane  four  directions  at  one  0 
and  two  at  the  other.  These  could  not  be  connected  by  finite  inter- 
mediate parts  alone.  Hence  the  finite  loops  are  on  the  same  side  of 
00' .  There  are  two  intermediate  parts  in  this  half  plane  and  four 
in  the  other.  Six  forms  are  possible  ;  three  with  three  branches, 
two  with  two,  and  one  with  one  branch. 

C.  If  the  two  loops  at  0  are  on  the  same  of  00' ,  those  at  0'  are 
also.  This  requires  four  intermediate  parts  on  the  other  side  of  00' . 
There  are  three  forms,  one  with  one  branch  and  two  with  two 
branches.  If  the  two  loops  at  0  (and  therefore  at  0')  are  on  differ- 
ent sides  of  00' ,  there  are  twenty-seven  forms ;  four  with  one 
branch,  thirteen  with  two  branches,  six  with  three,  and  four  with 
four. 

*  In  this  discussion,  only  those  branches  are  considered  which  contain  one  or 
both  multiple  points.     If  other  branches  exist,  they  must  be  simple  ovals. 


b  PLANE    CURVES    OF    THE    EIGHTH    ORDER. 

D.  On  one  side  of  00'  there  are  two  loops  at  each  0;  and  on 
the  other  side  of  the  line  there  are  single  loops  at  each  0  and  two 
intermediate  parts.  Twenty-one  forms  are  possible ;  six  with  one 
branch,  nine  with  two  branches,  and  six  with  three. 

E.  With  four  loops  at  each  0,  intermediate  parts  are  lacking. 
There  are  seven  forms ;  two  with  two  branches,  two  with  three,  and 
three  with  four. 

In  class  I  there  are  sixty-five  possibilities ;  twelve  with  one 
branch,  twenty-eight  with  two  branches,  seventeen  with  three,  and 
eight  with  four. 

In  classes  II  and  III,  curves  cannot  be  projected  entirely  into  the 
finite  portion  of  the  plane. 

Class  II. — A.  There  are  no  infinite  loops  but  two  infinite  interme- 
diate parts. 

1.  If  there  are  no  loops,  there  are  six  finite  intermediate  parts. 
Duly  one  arrangement  of  these  is  possible.     It  has  three  branches. 

2.  If  the  two  finite  loops  are  on  the  same  side  of  the  line  00' , 
only  one  of  the  four  intermediate  parts  lies  in  this  half  plane. 
There  are  six  forms ;  two  with  one  branch,  and  four  with  two 
branches.  If  the  loops  are  on  different  sides  of  00' ,  there  are  two 
finite  intermediate  parts  in  each  half  plane.  Six  forms  appear ; 
three  unipartite,  two  bipartite,  and  one  tripartite. 

3.  If  the  two  directions  to  infinity  from  0 (and  therefore  from  0) 
are  on  different  sides  of  00',  nine  forms  have  one  branch,  ten  hav^e 
two  branches,  and  eight  three.  If  both  directions  to  infinity  from  O 
are  in  one  half  plane,  those  at  0'  are  in  the  other  half  plane.  Six 
forms  are  unipartite,  nine  bipartite,  and  three  tripartite. 

B.  One  loop  and  one  intermediate  part  are  infinite. 

1.  The  one  finite  loop  at  0'  must  be  on  the  same  side  of  00'  as 
two  of  the  directions  to  infinity  from  0.  Hence  there  are  two  finite 
intermediate  parts  in  this  half  plane  and  three  in  the  other.  There 
are  three  forms,  two  bipartite  and  one  unipartite. 

2.  If  the  two  finite  loops  at  0'  are  on  the  same  side  of  00',  the  one 
at  0  is  also  in  this  half  plane.  The  finite  intermediate  parts  are  in 
the  other  half  plane.  There  are  six  forms,  one  has  one  branch,  three 
have  two  branches  and  two  three  branches.  If  the  two  finite  loops 
at  0'  are  on  different  sides  of  00',  only  one  of  the  three  finite  inter- 


PLANE    CURVES    OF    THE    EIGHTH    ORDER.  7 

mediate  parts  is  in  the  same  half  plane  as  the  one  finite  loop  at  0. 
Twenty-seven  forms  appear,  of  which  three  can  be  rejected  at  once, 
because  it  is  possible  to  construct  Hues  to  cut  the  curves  in  more 
than  eight  points.  Of  the  twenty-four  forms  remaining,  six  have  one 
branch,  twelve  have  two  branches,  and  six  have  three. 

3.  Since  there  are  two  directions  to  infinity  from  0  in  one  half  plane, 
the  two  finite  loops  at  0  are  on  different  sides  of  00'.  There  are  fifty- 
four  forms ;  six  have  one  branch,  eighteen  two  branches,  twenty-one 
three,  and  nine  four. 

C  There  are  two  infinite  loops.  (Hence  there  are  two  directions 
to  infinity  from  0  in  each  half  plane.) 

1.  If  there  are  no  finite  loops  at  0,  there  are  two  finite  inter- 
mediate parts  on  each  side  of  00' .  Hence  the  two  finite  loops  at  G 
are  in  different  half  planes.  Of  the  six  possible  arrangements,  four 
are  rejected  because  lines  can  be  found  to  cut  the  curves  in  more  than 
eight  points.  One  of  the  remaining  forms  is  unipartite  and  the  other 
is  bipartite. 

2.  If  there  is  one  finite  loop  at  0,  it  is  in  the  same  half  plane  as 
two  of  the  finite  loops  at  0',  for  there  can  be  no  finite  intermediate 
parts  in  this  half  plane.  There  are  two  in  the  other  half  plane.  Of 
the  eighteen  arrangements,  nine  are  rejected  for  the  reason  given  in 
1.  Two  of  the  remaining  forms  have  one  branch,  three  have  two 
branches,  three  have  three,  and  one  has  four. 

3.  Since  there  are  four  loops  at  0,  there  can  be  no  intermediate 
parts.  Twenty-one  arrangements  appear,  of  which  six  are  rejected 
for  the  reason  given  in  1.  Of  the  fifteen  curves  remaining,  two  have 
two  branches,  six  have  three,  five  have  four,  and  two  have  five.  It 
is  interesting  to  note  that  these  are  the  only  examples  of  these  curves 
with  more  than  four  branches. 

Class  III.  —  (Since  a  +  cZ  =  4,  6  =  0.) 

A,  Since  the  four  infinite  portions  are  intermediate  parts,  there 
are  no  loops  at  0.  There  is  only  one  arrangement  possible  for  the 
four  finite  intermediate  parts.     The  curve  is  bipartite. 

B.  If  one  of  the  four  infinite  parts  is  a  loop,  there  is  a  finite  loop 
at  0' .  It  is  in  the  half  plane  which  has  three  directions  to  infinity 
from  0.  One  of  the  finite  intermediate  parts  is  in  this  half  plane. 
One  of  the  forms  is  tripartite  and  two  are  bipartite. 


8  PLANE    CURVES    OF    THE    EIGHTH    ORDER. 

C  If  there  are  two  infinite  loops  and  two  infinite  intermediate 
parts,  there  are  four  directions  to  infinity  from  0  in  one  half  plane. 
The  two  finite  loops  at  0'  must  lie  in  this  same  half  plane,  and  the 
two  finite  intermediates  in  the  other  half  plane.  There  are  two 
forms  having  three  and  four  branches  respectively. 

D.  If  three  of  the  infinite  parts  are  loops,  there  are  four  directions 
to  infinity  from  Q  in  one  half  plane.  Hence  two  of  the  three  finite 
loops  at  0  lie  in  this  half  plane.  There  are  six  arrangements,  but 
five  are  rejected  for  the  reason  given  above.  The  curve  has  four 
branches. 

E.  The  presence  of  four  infinite  loops  at  0  necessitates  four  finite 
loops  at  0' .  Two  arrangements  appear,  but  both  are  rejected  for 
the  reason  given  above. 

All  forms  with  infinite  branches  have  now  been  found.     Of  the 

two  hundred  and  seven,  twenty-nine  were  rejected  because  in  each 

case  a  line  could  be  found  to  cut  the  curve  in  more  than  eight  real 

points.     One  hundred  and  seventy-eight  curves  remain  ;  thirty-seven 

have   one   branch,   sixty-nine  have  two   branches,   fifty-three  have 

three,  seventeen  have  four,  and  two  have  five.     These,  with  the 

sixty-five   which   are   entirely   in    the   finite   portion   of  the   plane, 

include  all  possible  forms  of  these  curves.     Of  the  two  hundred  and 

forty-three,  ninety-nine  contain  odd  branches.     Thirty-seven  have 

two  odd  branches,  forty-three  have  two  odd  and  one  even,  fifteen 

have  two  odd  and  two  even,  two  have  two  odd  and  three  even,  and 

two  have  four  odd. 

2.  Existence. 

All  possible  forms  of  the  C^  have  been  described  ;  but  there  is  no 
assurance  that  these  curves  actually  exist.  The  question  of  their 
existence  is  now  before  us.  Let  C\,  C*,  C*,  C*  be  four  plane 
quartics  having  at  0  and  0'  nodes  with  distinct  tangents.  Then 
Cj  C*  +  ^^'3  ^'4  =  ^  represents  a  curve  of  the  eighth  order  with  four- 
fold points  at  0  and  0'.  If  X  is  very  small  the  curve  is  nearly 
C*Cl  =  0,  but  with  breaks  at  the  eight  points  of  intersection  of  C* 
and  C*  distinct  from  O  and  0' .  By  taking  for  C*  and  C*  all  the 
forms  of  binodal  quartics,  by  placing  them  in  all  possible  positions 
relative  to  one  another  (so  long  as  the  eight  outside  intersections  of 
C\  and  C^  are  distinct  from  those  of  C^  and   C^),  and  by  making 


PLANE    CURVES    OF    THE    EIGHTH    ORDER.  » 

breaks  at  these  eight  outside  points  of  intersection,  one  can  obtain 
various  forms.  Since  all  of  the  two  hundred  and  forty-three  forms 
are  obtainable  in  this  way,  these  possible  forms  actually  exist. 

3.  Classification. 

Since  only  fifty  of  the  two  hundred  and  forty-three  curves 
are  equivalent  to  others  by  homographic  transformations,  it  seems 
unwise  to  use  homographic  differences  as  the  basis  of  classification. 
By  a  suitable  birational  transformation,  a  C^  with  two  four-fold 
points  goes  into  another  C^^  with  two  four-fold  points  ;  but  one  could 
not  be  certain  that  every  four-fold  point  had  separate  tangents.  The 
method  of  classification  actually  used  is  a  modification  of  the  ideas 
expounded  by  Tait,*  Kirkman,t  Little,  |  and  others  and  applied  by 
F.  Meyer,  §  and  P.  Field  1|  to  certain  curves  of  the  fourth  and  fifth 
orders.  Briefly,  their  scheme  is  as  follows  :  The  symbol  for  a  curve 
indicates  the  order  in  which  a  moving  point  describing  the  curve 
goes  through  the  multiple  points.  Calling  the  multiple  points 
A,  B,  C,  D,  ■••  the  symbol  AABBCDCD  is  distinguished  from 
AABCDBCD.  Since  the  moving  point  may  start  at  any  point  of 
the  curve,  cyclic  permutations  of  the  letters  in  the  symbol  are  per- 
missible, i.  e.,  AABCDBCD  is  the  same  as  BCDBCDAA, 
BCDAABCD,  CDBCDAAB,  etc.  Two  curves  are  regarded  as 
equivalent  if  they  have  the  same  symbol.  Meyer  used  this  method 
for  unicursal  curves  only.  Field  has  used  it  for  unicursal  quintics 
with  distinct  double  points.  In  his  paper  on  quintic  curves  of  defi- 
ciency one,  he  uses  the  same  method  when  the  curves  are  unipartite. 
Two  bipartite  curves  are  regarded  as  equivalent  if  the  symbols  for 

*  Seven  articles  of  the  Proceedings  of  Edinburgh  Royal  Society,  Vol.  9,  1877.  One 
article  in  the  Transactions  of  Edin.  Roy.  Soc,  Vol.  28,  1877.  One  article  in  the 
Philosophical  Magazine,  Vol.  17,  1884.  Two  articles  in  the  Trans.  Edin.  Roy.  Soc, 
Vol.  32,  1885. 

fTwo  articles  in  the  Trans.  Edin.  Roy.  Soc,  Vol.  32,  1885.  Two  articles  in  the 
Proc  Edin.  Roy.  Soc,  Vol.  13,  1886. 

JOne  article  in  the  Trans.  Edin.  Roy.  Soc,  Vol.  35,  1889.  One  article  in  the 
Trans.  Edin.  Roy.  Soc,  Vol.  36,  1890. 

§  Anwendungen  der  Topologie  auf  die  Gestalten  der  algebraischen  Curven.  Dis- 
sertation ;  Magdeburg,  1878.  Ueber  algebraische  Knoten,  Proc.  Edin.  Roy.  Soc, 
Vol.  13,  1886. 

II  On  the  Forms  of  Unicursal  Quintic  Curves,  American  Journal  of  Mathematics, 
Vol.  26.     Quintic  Curves  for  which  p^l,  American  Journal  of  Mathematics,  Vol.  27. 


10  PLANE  CURVES  OF  THE  EIGHTH  ORDER. 

the  two  parts  of  one  curve  are  the  same  as  those  for  the  separate 
parts  of  the  other.  Although  the  present  paper  employs  the  same 
method  for  unipartite  and  bipartite  curves  and  similar  schemes  for 
those  with  more  than  two  branches,  it  differs  from  these  other  papers 
in  applying  it  to  curves  with  four-fold  points. 

The  symbols  for  the  curv'es  under  discussion  must  contain  four  O's 
and  four  0''s.  For  convenience  in  writing,  represent  0  by  ^  and  0' 
by  B.  One  finds  that  there  are  seven  distinct  symbols  for  unipartite 
curves : 

1.  AAAABBBB. 

2.  AAABBBAB. 

3.  AAABBABB. 

4.  AABBAABB. 

5.  AABBABAB. 

6.  AABABBAB. 

7.  ABABABAB. 

For  the  first  six  symbols  it  is  actually  possible  to  get  corresponding 
unipartite  curves.  It  is  easily  shown  that  it  is  impossible  to  get  a 
unipartite  curve  of  symbol  (7)  without  introducing  additional  double 
points.  The  forty-nine  curves  of  one  branch  are  reduced  to  six 
classes,  denoted  by  the  first  six  symbols. 

In  the  following  tables  the  curves  corresponding  to  each  symbol 
are  given  by  their  numbers.  In  the  plates  only  one  picture  is  given 
for  two  curves  that  can  be  obtained  from  one  another  by  homographic 
transformations.  These  curves  are  indicated  by  *.  The  number  of 
curves  in  the  plates  is  therefore  one  hundred  and  ninety-three, 
instead  of  two  hundred  and  forty-three ;  since  fifty  can  be  obtained 
from  others  by  homographic  transformations. 

The  table  for  unipartite  curves  is  : 

1.  AAAABBBB.—   I  D  4,  6,  7,  12,  15,  19. 

II  5(3)  2',  16*,  18*. 
II  C(2)  2,  8. 

2.  AAABBBAB.  — 11  A  (3)  10*,  12*,  13*. 

II  B{2)  13,  14,  16,  18. 

3.  AAABBABB.—   1  CI,  10,  16,  18. 

II  A  (3)  17*,  19*. 
II  B  {2)  1,  4,  11. 


PLANE    CURVES    OF    THE    EIGHTH    ORDER.  11 

4.  AABBAABB.—   I  C  3. 

II  A  (3)  1*,  6,  22*. 
II  (7(1)  1. 

5.  AABBABAB.  —  ll  A  (2)  4,  5,  10, 

II  B  (1)  2. 

6.  AABABBAB.—   I  B  4. 

II  ^  (2)  7,  9. 

There  are  ninety-seven  forms  of  bipartite  curves  and  they  fall  into 
seventeen  classes.  A  dash  is  used  to  separate  the  symbols  for  the 
two  branches. 

1.  A-AABBABB.—  II  B  (2)  2,  3'. 

2.  A-AAABBBB.—   II  B  (3)  10",  24",  26*. 

II  C  (2)  5,  9. 

3.  A-AABBBAB.—  II  B  (2)  21,  22,  24,  26. 

4.  ^-^^^^5^5.-111  5  r. 

5.  AAA-ABBBB.—   II  ^  (3)  5*,  19*,  21*. 

6.  AAB-AABBB.—  II  A  (3)  2*,  5*,  23",  25*. 

11^(2)12,19,1'. 
II  (7(1)  2. 

7.  AAB-ABABB.—  II  ^  (2)  6,  8. 

II  B{1)  3. 

8.  AA-ABABBB.—   II  ^  (3)  8*,  IT,  14*. 

II  B  (2)  15,  17. 

9.  AA-AABBBB.—     I  Z>  2,  5,  8,  11,  13,  16,  17,  18,  21. 

11^(3)  r,3*,  17*. 
11(7(2)1. 

10.  AA-ABBABB.—     I  (7  5,  8,  11,  17. 

II  A  (3)  18,  20,  21. 

11.  AB-AABBAB.—   II  A  (2)  2,  12. 

I  ^  5,  6. 
115(1)  1. 

12.  AB-AAABBB.—     I  (7  19,  24,  27. 

II  B  (2)  5,  8. 
IS.  AAAA-BBBB.—     1  E  2\ 

II  (7  (3)  7*. 


12  PLANE  CURVES  OF  THE  EIGHTH  ORDER. 

14.  AAAB-ABBB.—     I  (7  20,  23,  28. 

II  B  (2)  6,  10. 

15.  AABB-ABAB.—  II  A  (2)  1,  3. 

16.  ^^^^-^^^^.  — III  A  1. 

17.  AABB-AABB.—     I  (7  1,  2,  4,  6,  15. 

II  A  (3)  16*. 

There  are  seventy  forms  of  curves  with  three  branches,  and  they 
are  in  sixteen  classes. 

1.  A-AA-ABBBB.—  II  B  (3)  8,  13,  15,  22*,  27*,  29*. 

2.  A-BB-AAABB.—  II  B  (3)  9*,  11*,  25*. 

II  C  (2)  4. 

3.  A-BB-ABAAB.—  II  B  (2)  23,  25. 

4.  A-ABB-AABB.—   II  5  (2)  3,  20,  27,  2'. 

5.  ^- J 5^-^ 5^ 5.  — Ill  B  2. 

6.  A-AAA-BBBB.—   II  (7(3)  4*,  5*. 

7.  AAA-ABB-BB.—   II  ^  (3)  4*,  6*,  20*. 

8.  AAB-AAB-BB.—   11  A  (3)  3*,  4*,  24,  26,  27. 

9.  A-A-AABBBB.—   II  6^(2)  3,  7. 

10.  A-A-ABBABB.  —  IU  (7  1. 

11.  AA-AB-ABBB.—     I  C  12,  14,  21,  22,  25,  26. 

II  B  (2)  7,  9. 

12.  AA-BB-AABB.—     I  D  1,  3,  9,  10,  14,  20. 

13.  AA-BB-ABAB.—   II  A  (3)  7,  9,  15*. 

14.  AB-AB-ABAB.—  II  .4  (1)  1. 

15.  AB-AB-AABB.—     I  B  1,  2,  3. 

II  A  (2)  11. 

IQ.  AA-AA-BBBB.—     1  E  4,  Q. 

II  C  (3)  6*. 

There  are  twenty-five  forms  with  four  branches  and  they  are  in 
nine  classes. 

1.  AB-AB-AB-AB.—     1  A  U 

2.  AA-BB-AB-AB.—      I  C9,  13,  29,  30. 


PLANE    CURVES   OF   THE    EIGHTH    ORDER.  13 

3.  AA-BB-AA-BB.—     IE  1,2,,  5. 

4.  A-ABB-AA-BB.—   TI  B  (3)  T,  12*,  14*,  23,  28,  30. 

5.  A-A-AABB-BB.—   II  C(2)  6. 

6.  A-AAA-BB-BB.—   II  C  (3)  3*. 

7.  A-A-AA-BBBB.—  II  C  (3)  2,  9*. 

8.  ^-^-^5^-^55.-111  C2. 

9.  ^-^-^-^5555.  — IIIi>l. 

There  are  ouly  two  curves  with  five  branches  and  they  have  the 
same  symbol :  A-A-AA-BB-BB.    These  curves  are  II  C(3)  1,  8. 

In  the  plates  different  kinds  of  lines  are  used  to  distinguish  the 

different  branches  :  heavy  lines  ,  dots  .  .  .  . ,  dashes ,  dots 

separated  by  dashes  . .  _ .,  and  two  dots  separated  by  a  dash 


4.    Relation  to  Space  Curves. 

If  the  complete  intersection  C^  of  two  surfaces  S"^  and  iS*  of 
second  and  fourth  order  be  projected  on  a  plane  tt  from  a  point  A 
on  aS'^  but  not  on  C^,  the  projection  is  a  C^  with  four-fold  points  at 
O  and  O' ,  the  points  in  which  the  generators  of  S'^  through  A  meet 
the  plane  tt.  If  /S^  and  S^  do  not  touch  (i.  e.,  if  there  are  no  point 
singularities  on  C^),  there  are  no  point  singularities  on  C^,  except 
the  two  four-fold  points.  Conversely,  such  a  Cp  can  always  be 
considered  as  the  projection  of  some  C^  from  some  point  on  S"^  but 
not  on  *S'^* 

If  0  and  O'  are  real  points,  /S"  is  a  hyperboloid  of  one   sheet. 

If  the  generators  of  S^  through  O  and  0'  are  taken  as  edges  of  the 

tetrahedron    of  reference    (A,    0,    0' ,    F),   the   equation  of  S^  is 

a?jX^  —  a'2.r3  =  0  (1).     Every  point  on  C'^  satisfies  equation  (1)  and 

the  equation  of  S*.     By  taking  as  triangle  of  reference  in  tt  00' A' 

(where  A'  is  the  intersection  of  ^i^  with  tt),  a  correspondence  can 

be  set  up  between  the  points  x  of  S*  and  the  points  ?/  of  tt  by  the 

equations   rx^  =  y\,    rx^  =  y,y,_,    rx^  =  y,y^,    rx^  =y^^  (2).     Since  a 

point  X  of  CI  is  connected  with  a  point  y  of  (7^  by  equations  (2), 

the  equation  of  C^  expressed   in  .r's  gives  the  equation  of  S^.     The 

*  See  Clebsch,  Vorlesungen  ueber  Geometrie,  Vol.  II,  pp.  414  sq.  for  the  general 
theory. 


14  PLANE  CURVES  OF  THE  EIGHTH  ORDER. 

equation  of  CI  is  Zc,j,2/f  "'"'2/22/3  =  ^  («+  {&  or  c}^4).     In  full 
this  is :  a,y\  +  y\{ajj^  +  a^^)  +  y\{a^l  +  a^l  +  a^^y^)  +  yl{a^yl 

+     «82/3     +     %ylh    +    «103/22/D    +    2/l(«ll2/2    +    «123/3    +    «132/22/3     +     ^uMs 

+  «i5  2/j2/D  +  y\{(^ny\yz  +  ^172/22/1  +  «i3  2/2  2/I  +  (f'x,y\yl)  +  2/1  Ko  2/22/3 

+  «2l2/22/3   +  "222/22/3)  +  2/,(«232/22/3   +   «242/22/3)   +  ^252/22/3  =  ^   (3)' 

In  obtaining  the  equation  of  S*'  from  (3)  by  means  of  equations 
(2),  ten  terms  are  ambiguous  :  a^,  a^,  a^^,  a^^,  a^^,  a^^  (twice),  a,g,  a^^, 
«22-  For  y\y^y^  can  be  rendered  x\x^  or  x\x^x^,  y\yly^  as  x]x^x^  or 
a^jar^Xg,   etc.     The  equation   of  S*^   may  be  written  :  a^x\ -{- a^x\x^ 

^~    (.t'o't^i  t^^o     ~T~    ^4*^1  *^o    ^1       ^c*^!   *^o     ~|  fi       1        4    ^1       (*■- *C.  •*/,     ^~     Lt'QU/,*tr„    — ^    Ct-Q  JT.  ily-io . 

"""  ^10^1**' 2^3   "T*   '^11^2  "I"    ^12^3      '     '^13'^'2^3     I     "l4**'2*3    "T   '^15'^2'*'3    "^   "l6'*'2*'^4 
~t~    ^'17**^3'^4      I*    ^ia'^i**^2      4    "•  19      1      3      4     "^        20      2      4    "*  21      3      4      •"        22      1      4 

"T"    ^i^^i     I      "24?'3*4      '     ^25^4  ^    ^' 

A  ten-fold  infinity  of  aS^'s  can  be  passed  through  a  given  C^. 
Since  there  are  thirty-four  non-homogeneous  constants  in  the  general 
equation  of  an  *S^*,  twenty-four  conditions  are  put  on  >S'*  by  making 
it  contain  a  given  C\. 

It  is  always  possible  to  pass  four  cones  through  a  C\  of  the  first 
species,  for  the  single  infinity  of  aS'-'s  that  can  be  passed  through  C* 
form  a  pencil  of  conicoids.  Hence,  any  such  C*  can  be  considered 
in  four  different  ways  as  the  intersection  of  a  cone  and  another  coni- 
coid.  In  the  case  of  a  C\,  where  the  8^  has  ten  degrees  of  freedom, 
a  cone  is  not  necessarily  contained  among  the  ten-fold  infinity  of 
aS'-'s  ;  for  it  requires  more  than  ten  conditions  to  make  certain  that 
the  8^  is  a  cone  of  the  fourth  order.  If  there  is  a  cone  among  the 
*S*'s  that  contain  C\,  the  following  theorems  hold  : 

Theorem  1.  The  curve  C\  is  the  envelope  of  the  conies  in  which 
the  planes  tangent  to  8^  cut  /S^ 

Proof.  —  Any  plane  through  a  generator  of  S^  cuts  S^  in  three 
other  generators.  Each  of  these  lines  has  two  points  on  (7^.  If  two 
generators  are  coincident  (i.  e.,  if  the  plane  is  tangent  to  S^),  two 
points  coincide  with  two  others  and  the  conic  is  doubly  tangent  to 
C^y.     Consecutive  tangent  planes  contain  consecutive  points  on  C^. 

Corollary.  —  The  curve  C^  is  the  envelope  of  the  projections  of 
the  conies  in  which  planes  tangent  to  8*  cut  8'^. 

Theorem  2.  —  The  surface  8^  can  be  pictured  by  a  single  infinity 
of  pencils  of  conies,  each  containing  C,  the  conic  in  which  tt  cuts  8"^. 


PLANE    CURVES   OF    THE    EIGHTH    ORDER.  15 

To  each  generator  of  S*  there  corresponds  a  pencil  whose  base  points 
are  0  and  0'  and  the  two  points  Q  and  Q'  in  which  C  is  cut  by  the 
polar  reciprocal  with  respect  to  S^  of  the  generator  of  *^S'^ 

Proof.  —  Take  as  tt  the  plane  of  projection,  the  polar  plane  with 
respect  to  S^  of  P  the  vertex  of  S*.  Call  the  pole  of  00'  with  re- 
spect to  C,  A'.  Use  as  the  center  of  projection,  A,  one  of  the  two 
points  in  which  A'P  cuts  S^.  Consider  as  the  picture  of  any  point 
X  on  aS'*,  the  projection  of  the  conic  in  which  8^  is  cut  by  the  polar 
plane  of  x  with  reference  to  S^.  Since  each  conic  on  S^  cuts  the  two 
generators  through  A  (AO  and  A 0'),  each  conic  on  S^  is  projected 
into  a  conic  on  tt  through  0  and  0'.  The  picture  of  each  point  on 
S^  is  a  conic  in  tt  through  0  and  O'.  The  converse  is  not  true. 
Since  there  are  a  double  infinity  of  points  on  /S*  and  a  triple  infinity 
of  conies  in  tt  through  O  and  O',  there  must  be  another  condition  on 
the  conies  which  picture  points  on  S*.  It  will  be  shown  that  each 
generator  of  ;8'*  is  pictured  by  a  pencil  of  conies  whose  base  points 
are  O  and  0'  and  the  two  points  Q  and  Q'  in  which  C  is  cut  by  the 
polar  reciprocal  with  respect  to  S^  of  the  generator  of  S^.  The  pic- 
ture of  P  is  C,  the  conic  in  which  tt  cuts  S^.  Each  generator  of  S* 
cuts  a,  the  plane  tangent  to  S^  at  A,  in  a  point  T  whose  polar  plane 
T  contains  A.  But  every  plane  r  through  A  cuts  /6'^  in  a  conic 
which  projects  into  a  line  pair  composed  of  00' and  the  line  QQ'  in 
which  the  plane  t  cuts  tt.  Since  QQ'  is  conjugate  to  PT  with  ref- 
erence to  S',  the  polar  plane  of  any  point  on  PI  contains  QQ .  Since 
Q  and  Q'  are  on  tt  as  well  as  on  S',  the  pictures  of  the  points  on 
PT  contain  Q  and  Q'.  Therefore  a  generator  PT  is  pictured  by  a 
pencil  0,  O',  Q,  Q'.  Also,  a  pencil  with  two  base  points  at  0  and 
0'  and  two  others  at  the  two  points  in  which  Ois  cut  by  a  line  con- 
jugate to  a  generator  of  S^  is  the  picture  of  that  generator  of  S^. 

Corollary  1.  —  The  picture  of  the  C*  in  which  S^  is  cut  by  a  the 
plane  tangent  to  S'^  at  JL  is  a  curve  of  the  fourth  class  D. 

As  point  T  moves  along  the  curve  of  the  fourth  order  C^  in  which 
a  cuts  S*,  the  line  QQ'  envelopes  a  curve  of  the  fourth  class  D. 

Corollary  2.  —  The  curve  D  is  the  polar  reciprocal  with  respect  to 
C  of  D',  the  O*  in  which  8*  cuts  tt. 

The  polar  reciprocal  of  a  point  X  of  tt  with  respect  to  C  is  the 
intersection  of  tt  with  the  polar  plane  of  X  with  reference  to  8"^. 


16  PLANE  CURVES  OF  THE  EIGHTH  ORDER. 

If  A''  lies  on  U,  the  polar  plane  of  A"  contains  the  line  Q  Q'  which 
is  conjugate  to  PAT.  Hence  AT  has  this  line  QQ'  as  its  polar  with 
reference  to  C.  Conversely,  every  QQ'  is  the  polar  of  some  point 
X  on  D'  with  reference  to  C 

Corollary  3.  —  The  polar  reciprocal  QQ'  of  a  generator  PT  of  *S'* 
is  pictured  by  a  pencil  of  conies  through  O  and  O  and  B  and  B', 
the  projections  of  the  two  points  B^  and  B'^,  in  which  PT  cuts  S^.  . 
The  two  pencils  0,  O,  Q,  Q'  and  O,  0',  B,  B'  are  said  to  be  recip- 
rocal to  each  other. 

Since  the  conies  in  which  the  polar  planes  of  the  points  of  QQ'  cut 
S^  contain  B^  and  B'^,  their  projections  contain  B  and  B'.  The  pro- 
jection of  every  conic  on  S"^  contains  O  and  0'. 

Theorem  3.  —  The  locus  of  points  of  contact  of  tangents  from  a 
^oint  on  QQ'  to  conies  of  the  pencil  determined  by  O,  O',  Q,  Q'  is  a 
•conic  of  the  reciprocal  pencil.  This  point  is  the  pole  of  00'  with 
reference  to  this  conic. 

Proof.  —  Take  any  definite  point  Q^  on  QQ'.  The  locus  of  points 
of  tangency  of  tangents  to  aS'^  from  Q^  is  the  conic  K'  in  which  S^  is 
cut  by  the  polar  plane  of  ^j.  K'  projects  into  a  conic  K  of  the 
pencil  0,  O',  B,  B',  and  these  tangents  into  lines  from  Q^  to  points 
of  K.  These  lines  are  the  tangents  from  Q^  to  the  conies  of  the 
pencil  0,  O,  Q,  Q'  and  the  points  of  tangency  are  on  K.  For,  each 
plane  through  QQ'  contains  two  of  the  tangents  to  aS'^,  which  are 
tangent  to  the  section  oi  S^  in  this  plane.  This  conic  projects  into 
a  conic  of  the  pencil,  with  the  projections  of  these  two  lines  as  tan- 
gents from  Q^  and  the  points  of  tangency  on  K.  Conversely,  every 
line  from  a  point  Q^  to  a  point  on  K  is  a  tangent  to  some  conic  of 
the  pencil  O,  O',  Q,  Q'. 

Corollary  1.  —  Conversely,  every  conic  of  the  reciprocal  pencil  is 
the  locus  of  the  points  of  tangency  of  the  tangents  from  some  })oint 
on  QQ'  to  the  conies  of  the  pencil  O,  0',  Q,  Q'.  The  pole  of  00' 
with  reference  to  any  conic  of  the  pencil  O,  0',  B,  B'  lies  on  QQ'. 
Corollary  2.  —  The  line  QQ'  is  the  locus  of  the  poles  of  00'  with 
respect  to  the  conies  of  the  pencil  O,  O,  B,  B' .  The  line  BB'  is 
the  locus  of  poles  for  the  pencil  O,  O',  Q,  Q'. 

Theorem  4.  —  The  curve  D  is  the  locus  of  the  poles  with  respect 
to  S'^  of  the  planes  tangent  to  S*'. 


PLANE  CURVES  OF  THE  EIGHTH  ORDER.  17 

Proof.  —  Since  D  lies  in  tt,  tlie  polar  plane  of  any  point  of  D  con- 
tains P.  The  polar  of  any  point  on  D  with  respect  to  C  is  a  tan- 
gent to  D' ,  since  D  is  the  polar  reciprocal  of  D' .  Since  the  polar 
plane  of  any  point  on  D  contains  P  and  a  tangent  to  Z)',  such  a 
plane  is  tangent  to  S^.  Conversely,  every  plane  tangent  io  S^  has 
its  pole  on  D. 

Corollary.  —  The  curve  D  is  the  locus  of  the  poles  of  00'  with 
reference  to  those  conies  in  tt  which  are  the  projections  of  sections  of 
S'^  cut  out  by  planes  tangent  to  S*. 

The  polar  plane  of  any  point  Q^  on  line  QQ'  cuts  S"^  in  a  conic 
whose  projection  contains  0,  0',  B,  B' .  Q^  is  the  pole  of  00' 
with  reference  to  this  conic.  When  and  only  when  Q^  is  on  D,  the 
polar  plane  is  tangent  to  S*. 

Theorem  5.  —  If  conies  of  the  pencils  O,  O' ,  B,  B'  and  O,  0' , 
Q,  Q'  intersect  at  the  point  P,  the  tangents  are  harmonically  sepa- 
rated by  POand  PO'. 

Proof.  —  Let  Qp  be  the  pole  of  00'  with  reference  to  the  first 
conic  and  B^  the  pole  for  the  second.  By  theorem  3,  the  tangent 
to  the  first  conic  at  P  contains  B^  and  the  tangent  to  the  second  at 
P  contains  Q^.  Call  the  point  in  which  PQ,^  cuts  00',  ^and  let 
-S''  be  the  point  of  intersection  of  PB^^  and  00' .  The  pole  of  PQ^^ 
lies  on  00'.  Since  PIC'  is  the  tangent  to  the  first  conic  at  P,  the 
pole  of  PQ(,  is  on  PK' .  Therefore  K'  is  the  pole  of  PQ^  with 
respect  to  the  first  conic.  Hence  O  and  O'  are  harmonically  sepa- 
rated by  ^and  K' .  But  the  cross  ratio  of  this  range  equals  that  of 
the  pencil  determined  by  the  tangents  at  i?  and  the  lines  PO  and 
PO'. 

Theorem  6.  —  The  curve  C\  is  the  envelope  of  conies  through  O 
and  O'  which  possess  the  two  following  properties  :  1)  the  pole  of 
00'  with  respect  to  the  variable  conic  lies  on  a  certain  curve  of  the 
fourth  class  D ;  2)  the  variable  conic  and  a  certain  fixed  conic  ( C) 
through  O  and  O'  are  harmonically  separated  at  a  point  of  intersec- 
tion by  the  lines  to  0  and  O' . 

Proof.  —  By  the  corollary  to  theorem  1,  C^  is  the  envelope  of  the 
projections  of  the  conies  in  which  the  planes  tangent  to  S*'  cut  S'^. 
It  remains  to  show  that  such  conies  possess  the  two  properties  in 
question.     The  poles  of  00'  with  respect  to  such  conies  are  on  D 


18  PLANE  CURVES  OF  THE  EIGHTH  ORDER. 

(corollary  to  theorem  4).  To  obtain  the  second  condition,  recall  that 
such  conies  belong  to  the  pencil  O,  O',  B,  B'  and  that  the  fixed 
conic  C  belongs  to  every  pencil  0,  O',  Q,  Q' .  Then  by  theorem  5, 
the  conies  in  question  fulfil  the  second  condition.  The  converse 
obviously  follows  and  completes  the  demonstration. 

Theorem  7.  —  The  number  of  conies  of  this  system  which  touch 
any  line  of  tt  equals  twice  the  order  of  D. 

Proof,  —  Any  line  in  the  plane  determined  by  point  A  and  any 
line  a;  of  TT  projects  into  the  line  x.  Therefore,  any  conic  K  on  S^ 
which  touches  any  line  y  of  plane  Ax  projects  into  a  conic  tangent  to 
X.  Since  y  is  tangent  to  K,  it  is  tangent  to  S'^ ;  and  since  y  lies  in 
plane  Ax,  it  is  tangent  to  G,  the  conic  in  which  the  plane  Ax  cuts 
S^.  Since  the  only  conies  which  belong  to  the  system  are  projec- 
tions of  conies  on  S"^  cut  out  by  planes  tangent  to  S*,  the  question 
becomes,  How  many  tangents  to  G  lie  in  planes  tangent  to  S^'i 
Since  a  plane  tangent  to  S*  is  determined  by  P  and  a  line  tangent 
to  a  plane  section  of  S*,  the  question  is.  How  many  common  tan- 
gents are  there  to  G  and  the  C*  in  which  plane  Ax  cuts  S^'i  The 
class  of  this  (7*  equals  the  class  of  Z>',  which  is  the  same  as  the 
order  of  D.  Since  G  is  of  the  second  class,  the  number  of  such 
tangents  is  2m,  where  m  is  the  order  of  D. 

Theorem  8.  —  The  number  of  these  enveloping  conies  which  pass 
through  any  point  of  tt  is  m.  These  conies  have  a  second  point  in 
common. 

Proof.  —  Take  any  point  Y^  in  tt  and  draw  AY^  Join  P  to  Y[, 
the  point  in  which  AY^  cuts  S^,  Call  the  point  in  which  PY[  cuts 
TT,  Y^.  Through  Y^,  m  tangents  to  JD'  can  be  drawn,  for  D  and  Z>' 
are  reciprocals.  The  m  planes  determined  by  P  and  these  tangents 
are  tangent  to  S*.  Each  plane  cuts  S^  in  a  conic  which  projects 
into  a  conic  of  the  enveloping  system.  Since  each  plane  contains 
the  line  PY^^Y[,  the  conic  on  S^  has  Y[  and  the  projection  on  tt 
passes  through  Y^.  These  conies  through  Y^  contain  Y^  the  projec- 
tion of  Fj  the  second  point  in  which  PY^  cuts  S^. 

Corollary  1.  —  If  Y^  is  on  C^,  Y^  is  on  B'. 

Corollary  2.  —  If  Y^  is  on  C  (not  at  O  or  0'),  Y[  coincides  with 
Yy  and  therefore  J'^  coincides  with  Y^ 

Corollary  3.  —  The  eight  points  of  intersection  of  C  and  B'  are 

one;. 


PLANE   CURVES    OF   THE    EIGHTH    ORDER.  19 

Theorem  9.  —  The  tangents  to  C^  at  O  or  O'  are  the  tangents  to 
D  from  the  same  point. 

Proof.  —  If  J^j  is  at  O,  the  whole  line  AY^  lies  on  >S^  and  there- 
fore contains  four  points  Y[  on  (7^.  The  four  lines  joining  these 
points  to  P  lie  in  plane  PA  O  and  cut  tt  in  four  points  Y^  on  line 
A'O  (A'  is  the  projection  of  P).  By  the  first  corollary  to  the  last 
theorem,  these  four  points  1^  on  A'O  are  on  D'.  The  polars  of 
these  points  are  tangents  to  D  from  O.  Of  the  m  tangents  to  D' 
from  any  one  of  these  four  points  l^^,  the  tangent  to  D'  at  1^  counts 
as  two.  Each  of  the  m  planes  determined  by  P  and  one  of  these 
tangents  cuts  S'  in  a  conic  through  Y[.  The  tangent  to  such  a 
conic  at  Y[  lies  in  the  plane  tangent  to  S^  at  Y[.  Since  PY^  is  a 
generator  of  S*  and  Y[  is  in  a,  the  plane  tangent  to  S^  at  Y[  cuts  tt 
in  a  tangent  to  D.  Since  the  tangent  plane  at  Y[  contains  A,  every 
line  in  it  projects  into  this  tangent  to  D  from  0.  In  particular  the 
tangent  at  Y[  to  the  two  coincident  sections  cut  out  by  the  plane 
through  the  tangent  to  D'  at  Y^  projects  into  this  tangent  to  D  from 
O.  But  these  two  coincident  conies  are  the  limit  of  two  consecutive 
conies  of  the  system  and  hence  their  projection  touches  C^  at  O. 
Therefore,  a  tangent  to  D  from  O  is  a  tangent  to  C*  at  0. 

Corollary.  —  If  2)  contains  O  (or  0'),  two  or  more  tangents  to  C^ 
at  O  coincide. 

For  if  O  is  a  point  on  D,  at  least  two  of  the  tangents  to  D  from 
O  coincide. 

Theorem  10.  —  A  line  joining  O  or  O'  to  any  one  of  the  2m  points 
of  intersection  of  C  and  D  is  tangent  to  C^. 

Proof.  —  The  polar  plane  of  any  one  of  these  points  Y^  is  tangent 
to  /S*  (theorem  4)  and  S~.  Such  a  plane  cuts  S^  in  a  degenerate 
conic  whose  projection  is  the  line  pair  Y^O,  Y^O'.  Since  this  line 
pair  belongs  to  the  system  enveloping  (7^  (corollary  to  theorem  4), 
it  touches  C^,  twice.  Since  Y^  0  and  Y^  O'  are  projections  of  gen- 
erators of  S"^,  each  contains  four  points  of  C^  distinct  from  O  (or 
O).     One  point  of  tangency  is  on  OY^  and  the  other  is  on  O'Y^. 

Corollary.  —  The  system  of  conies  enveloping  C\  contains  2m  de- 
generate conies  consisting  of  a  line  through  0  and  another  through 

a. 

Theorem  11. — There  are  m  double  tangents  to  C^  concurrent  at 
A'. 


20  PLANE    CURVES    OF    THE    EIGHTH    ORDER. 

Proof.  —  The  polar  plane  of  any  one  of  the  m  points  of  intersection 
of  Z)  and  the  line  00  is  tangent  to  S^  (theorem  4),  and  contains 
the  line  A  A'.  Hence  the  conic  in  which  this  plane  cuts  S"^  projects 
into  a  degenerate  conic  of  the  system  consisting  of  the  line  0  O'  and 
a  tangent  AH  to  D'  from  A .     Each  conic  of  the  system  touches 

C^  twice.     Since  line  00'  cuts  four  distinct  branches  of  C^  at  0 

p  p 

and  four  others  at  O'  the  line  00'  can  not  touch  C^.  Hence  the 
line  AH  is  a  double  tangent  to  C^.  Since  the  line  00'  cuts  D  in 
m  points,  there  are  m  such  lines  AH. 

Theorem  12.  —  The  curve  C^  goes  into  itself  by  a  quadric  inver- 
sion whose  pole  is  A  and  whose  conic  is  C 

Proof.  —  The  points  B  and  P'  are  two  vertices  of  the  complete 
quadrangle  formed  by  O,  O',  Q,  Q'.  Since  C  is  a  conic  through 
these  last  four  points,  P  and  P'  are  harmonically  separated  by  the 
points  of  intersection  of  C  and  line  PP'.  The  points  P  and  P  are 
on  C^  and  the  line  PP'  contains  A.  On  each  line  through  A  there 
are  four  pairs  of  points  P,  P'  corresponding  to  the  four  generators 
of  S*  determined  by  the  four  points  in  which  this  line  cuts  P'. 
Since  the  points  of  a  pair  are  collinear  with  A  and  conjugate  to  C, 
each  goes  into  the  other  by  a  quadric  inversion  whose  pole  is  A  and 
whose  conic  is  C  Therefore,  C]  goes  into  itself  by  this  inversion. 
C^^  might  be  considered  as  an  anallagmatic  curve,  if  one  were  to 
extend  the  term  to  include  general  quadric  inversion,  instead  of 
limiting  it,  as  is  usually  done,  to  circular  inversion. 

Method  of  generating  a  C^  with  two  real  four-fold  points.  Let 
O  and  O'  be  any  two  real  points  on  any  conic  C  and  let  D'  be  any 
general  plane  curve  of  the  fourth  order  in  the  same  plane  as  C.  Let 
Q  and  Q'  be  the  two  points  in  which  C  is  cut  by  the  polar  with 
reference  to  C  of  any  point  on  P'.  Draw  the  lines  joining  Q  and 
Q'  to  O  and  0'.  The  double  points  P  and  P'  of  these  line  pairs 
are  on  a  C]  with  four-fold  points  at  0  and  O'. 

A  Contact  Transformation. 

By  a  contact  transformation  whose  characteristic  equation  is  r 
=  x^y  -\-  y^x  -}-  3  (7  -f  a;_j/  =  0,  a  certain  class  quartic  is  transformed 
into  a  C^^  with  two  four-fold  points. 

Proof.  —  Consider  the  order  cubic  composed  of  the  line  z  =  0  and 


PLANE    CURVES    OF    THE    EIGHTH    ORDER. 


21 


the  conic  C^  ■}-  xy  =  0,  having  y  =  0  tangent  at  ^  (1,  0,  0)  and 
X  =  0  tangent  at  B  {0,  1,  0).  The  first  polar  of  a  point  [x^,  y^,  z^) 
with  reference  to  the  cubic  z{Cz^ -{- xy)  ^  0  (1)  is  x^yz -\- y^zx 
4-  z^{3Cz'^  +  xy)  =  0(2').  Using  non-homogeneous  codrdiuates,  this 
becomes  r  =  x^y  -f  y^x  -{■  ZC  +  xy  =  0  (2).  This  can  be  used  as 
the  characteristic  equation  for  a  contact  transformation,  since* 


0 

dr 
dx 

dr 
dy 

dr 

av 

d\ 

dxy 

dxdx^ 

dydx^ 

dr 

av 

d'r 

%i 

dxdy^ 

^y^Vi 

0    2/1  +  2/    ^1  +  ^ 


y 


X 


0 


0 


=  x{y  +  y^x  +  2xy 
This  ={=  0,  because 
of  r  =  0. 


Corresponding  to  each  pair  of  values  (x^,  y^)  there  is  a  conic  which 
contains  A  and  B.  It  will  now  be  shown  that  every  conic  is  har- 
monically separated  from  the  fixed  conic  .ry  —  3(7=  0  (3)  by  the 
lines  to  A  and  B.  The  points  of  intersection  of  (3)  with  a  conic 
(aside  from  A  and  B)  are 


»'o  = 


3Cx^ 


-  3(7d=  \/9C'-Sx^y,C 


Vo  = 


_  -  3  (7=b  V9  C-  Sx^y^  C 


X, 


The  tangent  to  (3)  at  (x^,  y^  is  x^y  -\-  y^x  —  6  C  =  0  (4).  The  tan- 
gent to  the  r  conic  at  the  same  point  is  x[y^  +  y^  +  y  {qi\  -\-  x^ 
+  ^i2/o  +  *o3/i  +  6  (7  =  0  (5).  The  line  through  A  and  point  [x^,  y^ 
is  y  —  y^  =  0  (6).  The  line  joining  B  to  the  point  (x^,  y^)  is  x  —  cc^ 
=  0  (7).  Take  \^  so  that  (4)  =  y  —  y^  -^  X^(x  —  x^)  =  0  ;  and  X^ 
so  that  (5)  =  2/  —  y^  +  Xjx  —  x^)  =  0.     Then 


Vo       .        6  (7  -  jK^y, 


X,    :=   —  ,  '^1    = 


ic: 


2  >  ^^2 


2/1+2/0 


or 


\  =  - 


*^i  +  *^o 
2a'i2/o  +  a^o2/i  +  6  (7  +  2^,3/0 


^0(^1  +  ^0) 

Substituting  the  values  of  x^  and  y^, 

*Lie,  S.,  and  Scheffers,  G.,  Geometrie  der  Beruehrungstransformationen,  Vol. 
I,  p.  54. 


22  PLANE  CURVES  OF  THE  EIGHTH  ORDER. 


_  6  (7  =F  2 1/9  a-  3aji3/i  C  -  x,y^ 


1  x\ 


X  =  ±  2/9(7- 3a;,y^C- eg +  a;,y^  ^_-^ 


x' 


Therefore  the  cross  ratio  of  the  pencil  determined  by  lines  (4),  (5), 
(6),  and  (7)  equals  —  1 .  Hence  the  first  polar  of  any  point  of  the 
plane  with  respect  to  (1)  is  a  conic  containing  A  and  B  and  sepa- 
rated harmonically  from  (3)  by  lines  to  A  and  B.  Each  conic  has 
its  pole  of  AB  at  the  point  (—  x^,  —  y^,  -\-z^).  As  this  point  de- 
scribes a  general  curve  of  the  fourth  class,  the  conies  envelope  a  C^ 
with  quadruple  points  at  A  and  B.  The  curve  which  is  transformed 
into  C^  is  described  by  the  point  {-{-x^,  -f-  2/ij  +  ^J. 


LIFE. 

I,  Elizabeth  Buchanan  Cowley,  was  born  in  Allegheny,  Pennsyl- 
vania, where  I  received  my  early  education  in  the  public  schools.  I 
then  studied  at  the  Indiana  State  Normal  School  of  Pennsylvania  for 
two  years  and  received  the  degree  of  B.S.  in  July,  1893.  The  next 
four  years  were  spent  in  teaching  in  the  public  schools  of  that  state. 
I  was  given  a  life  diploma  by  the  state.  In  September,  1897,  I  en- 
tered Vassar  College  and  received  my  A.B.  in  June,  1901.  I  was 
awarded  the  graduate  scholarship  in  mathematics  and  astronomy  for 
the  next  year  and  obtained  my  A.M.  in  1902.  Through  the  courtesy 
of  Professor  Whitney  an  opportunity  was  given  to  me  to  work  out 
the  definitive  orbit  of  a  comet.  Dr.  Furness  gave  valuable  advice 
and  Miss  Whiteside  assisted  in  calculations.  In  1902  I  received 
an  appointment  as  instructor  in  mathematics  at  Vassar  College 
and  am  still  holding  this  position.  The  summer  vacations  of  1903, 
1904,  and  1905  were  spent  in  resident  study  at  the  University 
of  Chicago,  where  I  had  twelve  weeks  of  mathematics  and  physics 
each  year.  I  studied  under  Professors  Bolza,  Dickson,  Millikan, 
Moulton  and  Slaught  and  Doctors  Gale  and  Jewett  and  the  late  Pro- 
fessor Maschke.  In  February,  1906,  I  began  work  at  Columbia 
University,  attending  lectures  that  second  semester  and  the  entire 


PLANE  CURVES  OF  THE  EIGHTH  ORDER.  23 

year  from  September,  1906,  to  June,  1907.  I  also  took  the  grad- 
uate courses  offered  at  the  summer  school  in  1906.  My  work  was 
with  Professors  Kasner,  Keyser,  and  Maclay.  I  am  a  member  of 
the  American  Mathematical  Society,  the  Association  of  Teachers  of 
Mathematics,  and  Circolo  Matematico  di  Palermo.  The  Astrono- 
mische  Nachrichten  published  my  paper.  The  Definitive  Orbit  of 
the  Comet  1826II.  as  a  separate  pamphlet  at  Kiel,  Germany,  in 
1907.  I  have  also  published  shorter  articles  in  the  Bulletin  of 
the  American  Mathematical  Society.  I  desire  to  express  my  grati- 
tude to  all  my  teachers,  but  especially  to  Professor  Keyser,  who  is 
able  in  an  unusual  degree  to  inspire  his  students  with  a  hearty  en- 
thusiasm for  study.  To  Professor  Henry  S.  White,  as  head  of  my 
department  at  Vassar  College,  I  am  indebted  for  encouragement, 
criticism,  and  sympathy  with  my  work.  It  is  my  pleasure  also  to 
state  my  obligation  to  President  Taylor  and  the  trustees  of  this  col- 
lege for  permission  to  attend  lectures  at  Columbia  University  while 
teaching  at  Vassar  College. 


UNIV 

OF 


PLATE  II. 


10 


11 


12 


13 


/     V. 


19 


20 


•21 


22 


PLATE  III. 


25 


..    8 


PLATE  IV. 


10 


11 


12 


17    .. 


•.•■•■c» 


1  .-.  ; 


t 


IE 


PLATE  V. 


iia:(i) 


II  A  (2) 


II  A  (2) 
1 


10 


11 


PLATE  VI. 


PLATE  VII. 


II  A  (3) 
22 


23 


24 


25 


26 


PLATE  VIII. 


II  B  (2) 
12 


13 


14 


16 


10 


11 


19 


PLATE  IX. 


21 


22 


'-.^/:>  4 

-^/:-. 


*  m    •  m  •' 


24 


25 


26 


27 


PLATE  X. 


10 


11 


12 


^      'J 


21 


14 


23 


16 


PLATE  XI. 


26 


27 


28 


lie  (2) 


PLATE  XII. 


Cx.-; 


\ 


lie  (3) 
8 


lie  (3)  ^    J 


:#; 


■-'M-' 


\' 


7\-N 


\ 


"■^x/y 


N 


III  A 


\ 


IIIB 


iiie 


HID 


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<^M 


liM' 


m 


c 


»;>  f  -«-v  ^ 


